Approximation, Error, and Precedence for Physics Help
In physics, the numbers we work with are not always exact values. In fact, in experimental physics, numbers are rarely the neat, crisp, precise animals familiar to the mathematician. Usually, we must approximate. There are two ways of doing this: truncation (simpler but less accurate) and rounding (a little more difficult but more accurate).
The process of truncation deletes all the numerals to the right of a certain point in the decimal part of an expression. Some electronic calculators use this process to fit numbers within their displays. For example, the number 3.830175692803 can be shortened in steps as follows:
Rounding is the preferred method of rendering numbers in shortened form. In this process, when a given digit (call it r ) is deleted at the right-hand extreme of an expression, the digit q to its left (which becomes the new r after the old r is deleted) is not changed if 0 ≤ r ≤4. If 5 ≤ r ≤ 9, then q is increased by 1 (“rounded up”). Some electronic calculators use rounding rather than truncation. If rounding is used, the number 3.830175692803 can be shortened in steps as follows:
When physical quantities are measured, exactness is impossible. Errors occur because of imperfections in the instruments and in some cases because of human error too. Suppose that x a represents the actual value of a quantity to be measured. Let x m represent the measured value of that quantity, in the same units as x a . Then the absolute error D a (in the same units as x a ) is given by
D a = x m − x a
The proportional error D p is equal to the absolute error divided by the actual value of the quantity:
D p = (x m − x a )/ x a
The percentage error D % is equal to 100 times the proportional error expressed as a ratio:
D % = 100 ( x m − x a )/ x a
Error values and percentages are positive if x m > x a and negative if x m < x a . This means that if the measured value is too large, the error is positive, and if the measured value is too small, the error is negative.
Does something seem strange about the preceding formulas? Are you a little uneasy about them? If you aren’t, maybe you should be. Note that the denominators of all three equations contain the value x a , the actual value of the quantity under scrutiny—the value that we are admitting we do not know exactly because our measurement is imperfect! How can we calculate error based on formulas that contain a quantity subject to the very error in question? The answer is that we can only make a good guess at x a . This is done by taking several, perhaps even many, measurements, each with its own value x m1 , x m2 , x m3 , and so on, and then averaging them to get a good estimate of x a . This means that in the imperfect world of physical things, the extent of our uncertainty is uncertain!
The foregoing method of error calculation also can be used to determine the extent to which a single reading x m varies from a long-term average x a , where x a is derived from many readings taken over a period of time.
Practice problems for these concepts can be found at: Scientific Notation for Physics Practice Test
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